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Numerical Solution for Initial Value Problem Numerical Analysis.

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Presentation on theme: "Numerical Solution for Initial Value Problem Numerical Analysis."— Presentation transcript:

1 Numerical Solution for Initial Value Problem Numerical Analysis

2 Introduction x0x0 x1x1 x2x2 x3x3 xnxn

3 Well-posed Problem

4 Well-posed Formula: Example

5 Classes of Methods

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7 Euler Method

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9 Euler Method: Formula

10 Euler Method: Geometry Meaning

11 Euler Method: Example

12

13 Taylor order n

14 Taylor order n: Formula

15 Taylor order n: Example

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22 Interpolate other values

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25 Runge-Kutta Method (RK) Runge-Kutta 25

26 Taylor Theory in 2 Variables 26

27 Runge-Kutta order 2 27

28 Runge-Kutta order 2 28

29 Runge-Kutta order 2 29

30 Runge-Kutta order 2 30

31 RK2: Midpoint Formula 31

32 RK2: Modified Euler Formula 32

33 RK2: Heuns Formula 33

34 RK2: Example 34

35 RK2: Example 35

36 RK2: Example 36

37 RK2: Example 37

38 RK2: Example 38

39 Runge-Kutta order 4 39

40 RK4: Example 40

41 RK4: Example 41

42 RK4: Example 42

43 RK4: Example 43

44 RK4: Example 44

45 RK4 vs Other Methods 45

46 RK4 vs Other Methods 46

47 RK4 vs Other Methods 47

48 Multi-Step Method 48

49 Multi-Step Method 49

50 Multi-Step Method: Explicit case 50

51 Adams-Bashforth Formula 51

52 Multi-Step Method: Explicit case 52

53 Multi-Step Method: Explicit case 53

54 Multi-Step Method: Implicit case 54

55 Multi-Step Method: Implicit case 55

56 Predictor-Corrector Method 2 (Predictor Formula) (Corrector Formula) (inner iteration) 56

57 Predictor-Corrector Method 57

58 Predictor-Corrector Method 58

59 Predictor-Corrector Method: Example 59

60 Predictor-Corrector Method: Example 60

61 Predictor-Corrector Method: Example 61

62 Predictor-Corrector Method: Example 62

63 Predictor-Corrector Method: Example 63

64 High ODE and System of ODE

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68 RK4 in vector form

69 ABM4 in vector form

70 High ODE and System of ODE: Example

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79 Divided Different Method for BVP

80 Differential Equation to Divided Different Form

81 Divided Different Formula for Linear BVP

82 Existing of Solution

83 Divided Different for BVP: Example

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